Volume of Cone 

A cone is a three-dimensional geometrical figure with a circular base that narrows smoothly to an apex. Cones are one of the most fundamental geometric shapes that are used in mathematics, engineering, and everyday objects. Understanding the surface area and volume of cones is essential for students to grasp their real-life applications. This guide will explore the volume of cones and spheres and compare the volume of cones, cylinders, and spheres. 

1.0What is a Cone?

A cone is a 3-dimensional geometric shape with a circular base and a pointed edge on the top, called an apex. The line segment that joins the apex to the centre of the base is the height of the cone. Cones are commonly seen in everyday objects like ice cream cones, traffic cones, and funnels. Some of the sound properties of a cone are:

• A cone has a circular base.
• The apex of a cone is a pointed tip that is situated at the opposite end of the base.
• The perpendicular distance measured from the base to the apex is the height of the cone.
• The slant height of any given cone is the distance from the edge of the base to the apex.

2.0Volume of a Cone Formula

The volume of a cone is characterised by the amount of space it occupies. The standard volume of the cone formula is ⅓ 𝜋$r^2$h, where

Here, 

V represents the Volume of the cone,

r represents the Radius of its circular base, and

h represents the Height of the cone.

3.0Comparing the Volume of a Cone, Cylinder, and Sphere

The volume of the cone, cylinder, and sphere can be compared using their respective formulas. Refer to the table below for a better understanding of the volume of cones and spheres as well as cylinders: 

4.0Surface Area of a Cone

The surface area and volume of cones are two main concepts that every student needs to master. In addition to the volume of the cone formula, the surface area of cone is an important area that consists of the base area and the lateral surface area. Thus, the total surface area of the cone can be calculated with the formula:

A = πr (r + l)

Where A is the total surface area, r is the radius of the base, and l is the slant height. 

5.0Solved Examples of the Volume of Cone

Below are a few examples of the volume of cones that students must practice to grasp the concepts clearly. 

Example 1. Find the volume (V) of a cone with a radius (r) of 4cm and a height (h) of 9cm.

We all know that V= ⅓ πr²h

So, using the formula, we can find 

V= ⅓ x 3.1416 x 4² x 9; 

V= ⅓ x 3.1416 x 16 x 9; 

V ≈ 150.8 cm³.

Example 2: A company manufactures traffic cones with a base radius of 15 cm and a height of 45 cm. How much plastic material is needed per cone?

We all know that V= ⅓ πr²h

So, by putting all the values we find, 

V = ⅓ π x 15² x 45; 

V = ⅓ x 3.1416 x 225 x 45; 

V ≈ 10610.1 cm³.

Thus, each cone will take approximately 10610 cm³ of plastic material.

Example 3: A conical tent has a volume of 6.6 m³ and a base radius of 1 meter. Find the Slant height of the tent in meters.

Solution: Given that the radius of the tent is 1m and the volume of the tent is 6.6m³. Therefore: 

The volume of the tent (V) = ⅓ πr²h

$\begin{array}{rl}& 6.6=\frac{22\times 1^2\times h}{3\times 7}\\ & h=\frac{6.6\times 21}{22}\\ & h=6.3\mathrm{m}\\ & \text{ Slant Height }(l)=\sqrt{(6.3{)}^2+1^2}\\ & l=\sqrt{40.69}\\ & l\approx 6.38\mathrm{m}\end{array}$

Hence, the slant height of the tent is 6.38 m. 

Example 4: The volume of a cone is 750 cm³. If the height is twice the radius, find the radius and height of the cone.

Solution: Given, volume of a cone is 750 cm³, and h = 2r

volume of a cone = 750 cm³ 

$\begin{array}{rl}& \begin{array}{rl}& \frac{1}{3}\pi r^{2}h=750\\ & \frac{1}{3}\times \frac{22}{7}\times r^2\times 2r=750\\ & r^3=\frac{750\times 21}{22\times 2}\end{array}\\ & \begin{array}{rl}& r\approx \sqrt[3]{358.2}\\ & r\approx 7.1\mathrm{cm}\end{array}\end{array}$

Practice Problem Based on Volume of cone: 

• The volume of a conical flask is 565.2 cm³, and the height is 15 cm. What is the radius of the base?
• A cement company uses cone-shaped molds to shape small cement piles. If the radius of one mold is 10 cm and the height is 30 cm, how much cement (in cubic centimeters) is needed to fill one mold?
• A decorative cone is made of paper and has a Slant height of 5 cm with a base radius of 4 cm. Find the volume of the decorative cone.

6.0Surface Area and Volume of Cone Worksheet

Practising problems related to surface area and volume of cone worksheets is important for mastering the topic. Below are some sample problems that you can work on to grasp the concepts better:

1. Basic Calculation: Find the volume of a cone with a radius of 7 cm and a height of 10 cm.
2. Comparative Problem: A cylinder has the same base and height as a cone. How many times greater is its volume compared to the cone?
3. Real-Life Application: A birthday party hat is shaped like a cone with a height of 20 cm and a base diameter of 12 cm. Calculate the volume of air inside the hat.

7.0Conclusion

Understanding the volume of the cone formula is essential for students to grasp advanced mathematical concepts and real-world applications. By comparing it with the volume of a cone, cylinder, and sphere and practising relevant worksheets, students can master the concept. With practice, one can easily apply these principles in engineering, design, and daily life.